Soroosh Yazdani

Ternary Diophantine equations of signature (p, p, 3)

In this paper, we develop machinery to solve ternary Diophantine equations of the shape Axn + Byn = Cz3 for various choices of coefficients (A,B, C). As a byproduct of this, we show, if p is prime, that the equation xn + yn = pz3 has no solutions in coprime integers x and y with |xy| > 1 and prime n > p4p2 . The techniques employed enable us to classify all elliptic curves over Q with a rational 3-torsion point and good reduction outside the set {3, p}, for a fixed prime p.

Generalized Fermat equations: A miscellany

This paper is devoted to the generalized Fermat equation xp + yq = zr, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infinite families of generalized Fermat equations that appear amenable to current techniques. While the main tools we employ are based upon the modularity of Galois representations (as is indeed true with all previously solved infinite families), in a number of cases we are led via descent to appeal to a rather intricate combination of multi-Frey techniques.

LEVEL LOWERING MODULO PRIME POWERS AND TWISTED FERMAT EQUATIONS

We discuss a clean level lowering theorem modulo prime powers for weight

A Local Version of Szpiro’s Conjecture

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Multiplicative functions and k-automatic sequences

cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations

On the equation a³ + b³ⁿ = c²

ax^n+by^n+cz^n=0

Geometry of 'standoffs' in lattice models of the spatial Prisoner's Dilemma and Snowdrift games

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Fekete-like polynomials

Szpiros conjecture asserts the existence of an absolute constant K>6 such that if E is an elliptic curve over , the minimal discriminant Δ(E) of E is bounded above in modulus by the Kth power of the conductor N(E) of E. An immediate consequence of this is the existence of an absolute upper bound on min{vp (Δ(E)):p∣Δ(E)}. In this paper, we will prove this local version of Szpiros conjecture under the (admittedly strong) additional hypotheses that N(E) is divisible by a “large” prime p and that E possesses a nontrivial rational isogeny. We will also formulate a related conjecture that if true, we prove to be sharp. Our construction of families of curves for which min{vp (Δ(E)):p∣Δ(E)}⩾6 provides an alternative proof of a result of Masser on the sharpness of Szpiros conjecture. We close the paper by reporting on recent computations of examples of curves with large Szpiro ratio.